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Motives and the Pfaffian-Grassmannian equivalence. (English) Zbl 1487.14016

Summary: We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov’s conjecture for Borisov’s Calabi-Yau threefolds, and verifications of Kimura’s finite-dimensionality conjecture, Voevodsky’s smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
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